Unified Framework of Time--Geometry--Interaction:\\ From Scattering Phase Scale to Geometrization of Gravity and Gauge Forces

Unied F amewo k o TimeGeome yIn e ac ion:
F om Sca e ing Phase Scale o Geome iza ion o
G a i y and Gauge Fo ces
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
We cons uc a unied amewo k wi h  ime scale equi alence class as he co e
objec , ew i ing g a i y, gauge in e ac ions, and mac oscopic classical  o ces as
p ojec ions o he same geome ic s uc u e a die en hie a chical le els. Fi s , in
he igo ous con ex o sca e ing heo y, based on Bi manK en spec al shi unc-
ion and Wigne Smi h ime delay, we gi e he scale iden i y o phase de i a i e,
ela i e s a e densi y, and g oup delay ace: o a class o sca e ing sys ems sa is y-
ing ace-class pe u ba ion condi ions, we ha e
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
,
whe e
Q(ω) = −iS(ω)†∂ωS(ω)
is he Wigne Smi h ma ix,
ρ el
is he de i a i e
o K en spec al shi densi y. This iden i y unies phasedelay ela i e spec al
densi y as an obse able ime scale. Second, in oducing a o al bundle wi h space-
ime as base mani old and in e nal cha ge space and obse e esolu ion hie a chy
as be s, we p o e ha unde condi ions sa is ying local Lo en z and in e nal gauge
symme y, g a i y and YangMills gauge elds can be iewed as die en compo-
nen s o he same o al connec ion, whose cu a u es espec i ely gi e space ime
cu a u e and in e nal eld s eng h; he ime scale is ela ed o phase in eg a ion
along wo ldlines and pa allel anspo o he connec ion, hus de i ing he unied
p oposi ion ha no undamen al o ces, only ime cu a u e unde die en geome -
ic p ojec ions. Thi d, a he algeb aic quan um eld heo y le el, using modula
ow gi en by Tomi aTakesaki modula heo y as in e nal ime, combined wi h
quan um null ene gy condi ion (QNEC) and gene alized en opy mono onici y, we
cha ac e ize he a ow o ime as a pa ial o de s uc u e o ela i e en opy and
modula ene gy mono onici y. Finally, in he semiclassical limi , we gi e a heo em-
o m s a emen o  o ce = p ojec ed cu a u e o ime geome y o expec a ion
alue e olu ion o mac oscopic pa icles, explaining ha classical New onian me-
chanics, Lo en z o ce, and e en eec i e en opic o ce can all be iewed as eec-
i e desc ip ions o his unied ime geome y unde die en coa se-g aining and
in e nal s a e cons ain s. A he end, we p o ide se e al e iable enginee ing p o-
posals, including Wigne Smi h ime delay measu emen s in mic owa e ne wo ks
and mesoscopic conduc o s, a omic clock g a i a ional edshi phase scale econ-
cilia ion, and c oss-scale scale consis ency es s based on FRW cosmological edshi
and s ong g a i a ional lensing.
1
Keywo ds:
Time Scale Equi alence Class; Sca e ing Phase; Wigne Smi h Time De-
lay; Bi manK en Spec al Shi ; Geome iza ion o G a i y; YangMills Gauge Field;
Modula Flow; Quan um Null Ene gy Condi ion; Eme gence o Mac oscopic Fo ce

1 In oduc ion and His o ical Con ex
Classical physics akes  o ce as a undamen al concep : New on's second law
F=ma
iews o ce as he cause o changing a pa icle's eloci y. Field- heo e ic o mula ion o
elec omagne ism in some sense weakens he undamen al s a us o  o ce: Lo en z o ce
q(E+ ×B)
can be ob ained om con ac ion o elec omagne ic eld enso
Fµν
and
ou - eloci y
uν
, he eld i sel go e ned by Maxwell equa ions. Gene al ela i i y u he
upg ades g a i y om  o ce o space ime geome y: ee- all wo ldlines sa is y geodesic
equa ions, so-called g a i a ional accele a ion can be iewed as ine ial o ce caused by
choosing non- ee- all ames.
In mid- wen ie h cen u y, gauge heo y unied elec omagne ic, weak, and s ong in-
e ac ions as connec ions and cu a u es on p incipal bundles: gauge po en ial
Aµ
is
connec ion on be bundle, eld s eng h
Fµν
is cu a u e o his connec ion, co espond-
ing o Eule Lag ange equa ions o YangMills equa ions. Fo ce expe ienced by cha ges
in gauge elds can be unde s ood as ajec o y deec ion when doing pa allel anspo
in o al space wi h gauge connec ion. Geome iza ion o gauge heo y has shown: in e -
ac ions can be unied as die en ypes o geome ic s uc u es.
On he o he hand, quan um sca e ing heo y e eals p o ound connec ions among
phase, s a e densi y, and  ime delay. Bi manK en o mula es ablishes ela ionship
be ween sca e ing ma ix de e minan and spec al shi unc ion, F iedel sum ule con-
nec s phase and s a e densi y, Smi h and Wigne 's in oduced ime delay and Wigne 
Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
cha ac e ize ime s uc u e o sca e ing p ocess.
Subsequen wo k shows ha unde sui able ace-class pe u ba ion condi ions, phase
de i a i e, ela i e s a e densi y, and ace o Wigne Smi h ma ix uni y h ough igo -
ous ace o mulas.
In algeb aic quan um eld heo y, Tomi aTakesaki modula heo y endows any on
Neumann algeb a wi h a cyclicsepa a ing ec o wi h an in insic modula ow
σφ
.
Modula ow is gene a ed by one-pa ame e uni a y g oup
∆i
o modula ope a o
∆
,
and can be iewed as in e nal ime na u ally induced by s a ealgeb a pai . This
s uc u e plays an impo an ole in he mal ime hypo hesis and ime econs uc ion in
g a i yholog aphic backg ounds.
Recen quan um null ene gy condi ion (QNEC) wo k e eals a ow o ime s uc u e
be ween ene gyen opy: in qui e gene al quan um eld heo ies, s ess-ene gy enso
along null di ec ions sa ises an inequali y wi h second-o de a ia ion o on Neumann
en opy as lowe bound, hus binding geome ic changes o en opy wi h local ene gy
cons ain s.
These de elopmen s join ly poin o a deepe unied pic u e: Time is no jus an
ex e nal pa ame e , bu a scale join ly dened by mul iple s uc u es including sca e ing
phase, spec al shi , modula ow, and en opy cu a u e; G a i y and gauge in e ac ions
a e bo h connec ion cu a u es co esponding o die en be di ec ions; Mac oscopic
 o ces a e p ojec ions o his imegeome y a coa se-g aining and eec i e deg ees o
eedom le els.
2
The goal o his pape is o cons uc an explici unied amewo k based on he abo e
ma u e esul s:
1. Using phasespec al shi  ime delay iden i y in sca e ing heo y, gi e an obse -
able deni ion o ime scale;
2. Uni y g a i y and gauge in e ac ions as die en componen s o  ime connec ion,
de i ing mac oscopic o ce as p ojec ion o ime geome y cu a u e;
3. Use modula ow and QNEC o cha ac e ize a ow o ime and causal pa ial o de ;
4. P o ide e iable expe imen alenginee ing p oposals, making his unied ame-
wo k alsiable.

2 Model and Assump ions
2.1 Space ime and Quan um Field Theo y Backg ound
Le
(M, g)
be a ou -dimensional, ime-o ien ed globally hype bolic Lo en zian mani old
sa is ying o dina y causali y and ene gy condi ions. Physical sys em is gi en by quan um
eld heo y dened on
(M, g)
, wi h Hilbe space
H
, local obse able algeb a
A(O)⊂
B(H)
, obeying HaagKas le axioms. Choose e e ence s a e
Ω∈ H
(can be acuum o
he mal equilib ium), assuming i is cyclic and sepa a ing o app op ia e subalgeb as.
In cases wi h sucien ly asymp o ically a egions, assume exis ence o global ime
ansla ion symme y
7→ U( ) = e−iH
, co esponding gene a o
H
sel -adjoin , allowing
cons uc ion o sca e ing s a es and sca e ing ma ix.
2.2 Obse e , Resolu ion, and Coa se-G aining
Obse e is desc ibed by a imelike wo ldline
γ:τ7→ xµ(τ)
and a se o de ec o deg ees
o eedom. In oduce a  esolu ion scale pa ame e
Λ
, cha ac e izing he ene gy ime
ange and spacemomen um window his obse e can esol e. Fo mally, coa se-g aining
ope a o s can be iewed as amilies o comple ely posi i e, ace-p ese ing maps
{ΦΛ}
on algeb a
A(O)
; as
Λ
dec eases, e ained deg ees o eedom become coa se .
Rela ionship among esolu ion ime edshi is cha ac e ized in he unied amewo k
h ough equi alence class: die en
(γ, Λ)
combina ions can dene he same ime scale
(in he sense below).
2.3 Sca e ing Sys em and Scale Iden i y
Conside a pai o sel -adjoin ope a o s
H, H0
on
H
, sa is ying
V:= H−H0
is ace-class
pe u ba ion and sa is ying s anda d wa e ope a o exis ence and comple eness condi-
ions. Le
S(ω)
be xed-ene gy sca e ing ma ix, whose de e minan sa ises Bi man
K en o mula wi h spec al shi unc ion
ξ(ω;H, H0)
:
de S(ω) = e−2πiξ(ω)
almos e e ywhe e
.
3
Dene o al sca e ing phase
Φ(ω) = a g de S(ω)
, and se
φ(ω) = 1
2Φ(ω) = −πξ(ω).
In oduce Wigne Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω),
whose ace in many models can be in e p e ed as o al g oup delay.
Rela i e s a e densi y is dened as
ρ el(ω) = ρ(ω;H)−ρ(ω;H0),
whe e
ρ
is densi y o s a es. K en ace o mula gi es
( (H)− (H0)) = ZR
′(λ)ξ(λ) dλ,
app op ia ely choosing
can deduce
ρ el(ω) = ξ′(ω)
almos e e ywhe e.
Unde good in eg abili y condi ions, Wigne Smi h ma ix ace sa ises
Q(ω)=2∂ωφ(ω).
Combining yields he scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
almos e e ywhe e
.
This deni ion unies addi ional s a e densi y,  o al g oup delay, and sca e ing
phase g adien  as a single unc ion
κ(ω)
, which can be iewed as  ime scale densi y o
gi en sca e ing sys em and e e ence Hamil onian.
2.4 Geome iza ion: To al Bundle and Connec ion
Le
M
be space ime mani old, in oduce o al bundle
π:B → M,
whose be is he p oduc o in e nal cha ge space
Fin
and  esolu ion space
F es
. Fo
each poin
x∈M
, be
Bx
ep esen s in e nal deg ees o eedom o quan um s a e a
ha poin and measu emen window a ailable o obse e .
In oduce p incipal bundle s uc u e on
B
, whose s uc u e g oup is
G o =SO(1,3)↑×GYM ×G es,
espec i ely co esponding o local Lo en z g oup, in e nal YangMills gauge g oup, and
esolu ion scaling g oup. To al connec ion is w i en as
Ω=ωLC ⊕AYM ⊕Γ es,
whe e
ωLC
is Le iCi i a spin connec ion,
AYM
is s anda d YangMills gauge eld,
Γ es
desc ibes esolu ion ow in RG sense (e.g., eno maliza ion g oup o coa se-g aining con-
nec ion).
To al cu a u e
R= dΩ+Ω∧Ω
na u ally decomposes in o space ime cu a u e
R
, YangMills eld s eng h
F
, and es-
olu ion ow cu a u e
R es
.
4
2.5 Modula Time and En opy A ow
Fo each local obse able algeb a
A(O)
and s a e
φ
(gi ing ec o s a e om
Ω
), Tomi a
Takesaki heo y gi es modula ope a o
∆φ
and modula au omo phism g oup
σφ
(A) = ∆i
φA∆−i
φ.
Modula ow pa ame e
can be in e p e ed as in e nal ime o his obse e algeb a
pai ; in holog aphic and he mal ime hypo hesis con ex s, i is iewed as a candida e o
geome ic ime.
QNEC s a es ha o app op ia e classes o quan um eld heo ies and s a es,
⟨Tkk(x)⟩ψ≥1
2πS′′
N(x),
whe e
Tkk
is s ess-ene gy componen along null ec o
kµ
,
S′′
N
is second-o de a ia ion o
on Neumann en opy along ha di ec ion. This inequali y and mo e gene al gene alized
en opy mono onici y p o ide quan i a i e basis o dening a ow o ime and causal
pa ial o de om quan um in o ma ion pe spec i e.

3 Main Resul s (Theo ems and Alignmen s)
This sec ion gi es co e deni ions and heo ems o he unied amewo k, and indica es
alignmen ela ionships among hem.
3.1 Deni ion o Time Scale Equi alence Class
Deni ion 3.1
(Ope a ional Time Scale (Deni ion 3.1))
.
Gi en a sca e ing sys em
(H, H0)
and i s sca e ing ma ix
S(ω)
, dene scale densi y
κ(ω) = φ′(ω)/π,
whe e
φ(ω)
is de e mined by a o emen ioned Bi manK en ela ion. Fo ene gy window
I⊂R
, dene eec i e ime scale
τI(E) = ZE
E0
κ(ω)1I(ω) dω,
which expe imen ally co esponds o in eg al o addi ional g oup delay ob ained om
equencyphase measu emen s.
Deni ion 3.2
(Time Scale Equi alence Class (Deni ion 3.2))
.
Gi en wo se s o ope -
a ional ime pa ame e s
and
′
, i he e exis s ic ly mono onic
C1
unc ion
:R→R
and global posi i e cons an
c > 0
, such ha o all sca e ing expe imen s ealizable in
common domain, we ha e
′=c ( )
and all obse able phase die ences and g oup delays ollow consis en o de ing wi h
espec o
, ′
, hen
and
′
a e said o belong o he same ime scale equi alence class,
w i en
[ ]=[ ′]
.
5

P oposi ion 3.3
(P oposi ion 3.3)
.
Unde condi ions whe e scale iden i y holds, o
any sca e ing sys em sa is ying ace-class pe u ba ion condi ions and gi en e e ence
(H0)
, ime scales dened by die en p obe amilies ( equency windows, inciden channel
choices) con e ge o he same equi alence class
[τ]
in he sense o equi alence ela ion, i
and only i ene gy dependence o Wigne Smi h o al ime delay is con olled by a unied
geome ic s uc u e (such as same eec i e po en ial and bounda y condi ion class).
This p oposi ion ensu es ha unde sui able uni e sal p obe amilies, ime scale has
p obe independence, hus can be used o dene mac oscopic ime.
3.2 PhaseSpec al Shi Time Delay Scale Iden i y
Theo em 3.4
(Scale Iden i y (Theo em 3.4))
.
Le
H, H0
be sel -adjoin ope a o s,
H−H0
be ace-class pe u ba ion, sca e ing ma ix
S(ω)
exis s and is uni a y. Le spec al shi
unc ion
ξ(ω)
be dened by K en ace o mula, hen almos e e ywhe e
[ (H)− (H0)] = ZR
′(λ)ξ(λ) dλ
o all app op ia e
.
Dene
φ(ω) = −πξ(ω)
,
Q(ω) = −iS(ω)†∂ωS(ω)
, hen almos e e ywhe e
φ′(ω)
π=ξ′(ω) = ρ el(ω) = 1
2π Q(ω).
This heo em combines Bi manK en o mula, F iedel sum ule, and Wigne Smi h
deni ion, gi ing unica ion o phase, spec al shi , and ime delay.
Co olla y 3.5
(Spec al Deni ion o Time Scale (Co olla y 3.5))
.
Scale densi y
κ(ω)
can be equi alen ly dened as ela i e s a e densi y, o as no maliza ion o Wigne Smi h
ma ix ace, hus ha ing comple e spec alsca e ing exp ession.
3.3 Space imeIn e nal Space Unied Geome y and Geome iza-
ion o Fo ce
On o al bundle
B
, cu a u e
R
o o al connec ion
Ω
can be decomposed as
R=R⊕F⊕ R es.
Pa allel anspo along a ma e pa icle wo ldline
γ
is con olled by o al co a ian
de i a i e
Dτ=d
dτ+Ω(˙γ).
E olu ion o in insic deg ees o eedom (such as spin, colo cha ge) and esolu ion a i-
ables is de e mined by
F
and
R es
.
Theo em 3.6
(No Fundamen al Fo ce P oposi ion (Theo em 3.6))
.
In semiclassical
limi , o pa icle wi h mass
m
and in e nal cha ge
q
, expec a ion alue o cen e -o -mass
ajec o y
xµ(τ)
sa ises
mD2xµ
Dτ2=qFµν
dxν
dτ+ µ
es
6
whe e
D
is Le iCi i a co a ian de i a i e,
Fµν
is p ojec ion o YangMills eld
s eng h in co esponding ep esen a ion,
µ
es
is eec i e en opic o ce e m caused
by
R es
and s a een opy changes. In o he wo ds, g a i y, Lo en z o ce, and en opy-
d i en o ce in classical sense a e all p ojec ions o o al connec ion cu a u e on die en
beha io spaces, wi hou need o sepa a ely in oduce p imi i e concep o  undamen al
o ce.
P oo based on semiclassical p opaga ion o wa e packe and pa h in eg al ep esen-
a ion, de ails in Appendix C.
3.4 Time Scale and Recons uc ion o G a i a ional Geome y
Theo em 3.7
(F om Scale Iden i y o G a i a ional Redshi (Theo em 3.7))
.
Conside
space ime egion wi h s a ic Killing ec o
∂
, me ic can be w i en in s anda d s a ic
o m
g=−N2(x)d 2+hij(x)dxidxj.
Assume sca e ing p ocesses wi h ene gy localized in
I
exis in his egion, whose
Wigne Smi h o al g oup delay
Q(ω)
can be measu ed by dis an obse e . I scale
densi y
κ(ω)
wi hin
I
is app oxima ely ela ed o posi ion only h ough escaling by
N(x)
,
i.e.,
κ(ω;x) = N−1(x)κ∞(ω)
hen ime scale dened by dis an obse e and p ope ime scale dened by local ee
all belong o same equi alence class, hei a io gi ing g a i a ional edshi ac o
N(x)
.
P oo elies on equency conse a ion in s a ic space and ela ion o local ene gy
ωloc =N−1(x)ω
, see Appendix B.
Co olla y 3.8
(Co olla y 3.8)
.
Unde abo e se ing, g a i a ional po en ial can be iewed
as escaling pa e n o unied ime scale be ween die en spa ial poin s; g a i a ional
ime dila ion and Shapi o delay a e wo eadou me hods o same ime geome y unde
die en ope a ional deni ions.
3.5 Gauge In e ac ion as Condi ioned Time Scale
P oposi ion 3.9
(Cha ge-Dependen Addi ional G oup Delay (P oposi ion 3.9))
.
In
sca e ing sys ems wi h
U(1)
o mo e gene al YangMills gauge eld
Aµ
, phase de i a i e
die ence o sca e ing ma ices
Sρ(ω)
co esponding o die en in e nal cha ge ep e-
sen a ions
ρ
∆κρ,ρ′(ω) = 1
2π [Qρ(ω)−Qρ′(ω)]
in semiclassical limi is equi alen o de i a i e wi h espec o ene gy o Wilson line
phase die ence along classical ajec o y, hus co esponding o ime delay die ence
caused by Lo en z o ce.
This p oposi ion shows ha gauge o ce can be unde s ood as die ence in condi-
ioned ime scale pe cei ed by die en cha ge sec o s in same geome ic backg ound.
7
3.6 Modula Time, En opy Mono onici y, and A ow o Time
Theo em 3.10
(Modula FlowGeome ic Time Alignmen Theo em (Theo em 3.10))
.
In a class o quan um eld heo ies desc ibable h ough geome ic holog aphy, i local
modula ow
σφ
co esponds o ow o some Killing o app oxima e Killing ec o on
g a i y side, hen ollowing s a emen s a e equi alen :
1. Modula ow ime
belongs o same ime scale equi alence class as geome ic ime
τ
;
2. Mono onici y o expec a ion alue o modula Hamil onian and gene alized en opy
ex emali y condi ion join ly de i e local g a i a ional eld equa ions;
3. QNEC holds and sa u a es in co esponding null di ec ion.
This heo em abs ac s he idea in exis ing wo k ha gene alized en opy ex emali y
+ QNEC suces o de i e Eins ein equa ions, es a ing i as consis ency condi ion
be ween ime scale equi alence class and modula ow.
P oo skele on in Appendix D.

4 P oo s
This sec ion gi es p oo ideas o main esul s and se e al key s eps, lea ing echnical
de ails o appendices.
4.1 P oo Ou line o Theo em 3.4
Co e ool is K en spec al shi unc ion heo y and ace o mula o Wigne Smi h
ma ix.
1. K en ace o mula ensu es o ace-class pe u ba ion
V=H−H0∈S1
, he e
exis s unique
ξ(λ)∈L1(R)
such ha o sucien ly good
,
[ (H)− (H0)] = Z ′(λ)ξ(λ) dλ.
2. Bi manK en o mula connec s sca e ing ma ix de e minan and spec al shi
unc ion:
de S(ω) = e−2πiξ(ω),
almos e e ywhe e
.
3. Dene o al sca e ing phase
Φ(ω) = a g de S(ω)
, choose con inuous b anch such
ha
Φ(ω) = −2πξ(ω)
(igno ing in ege cons an ). Dene
φ(ω) = 1
2Φ(ω) = −πξ(ω)
,
hus
φ′(ω)/π =−ξ′(ω) = ρ el(ω)
.
4. Wigne Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
. Unde uni a i y o
S(ω)
,
Q(ω) =
−i∂ωln de S(ω)
. By Bi manK en o mula,
ln de S(ω) = −2πiξ(ω),
hus
Q(ω) = 2πξ′(ω)
.
5. Combining yields
φ′(ω)/π =ξ′(ω) = (2π)−1 Q(ω).
Key is con olling b anch and die en iabili y o
ln de S
, de ails in Appendix A.
8
4.2 P oo Ou line o Theo em 3.6
Adop wa e packe semiclassical limi and pa h in eg al me hod:
1. Le single-pa icle wa e packe ini ial s a e be desc ibed by WKB- ype wa e unc-
ion, whose phase is gi en by ac ion
S[γ] = Z(pµ˙xµ−H) dτ.
2. On o al bundle
B
, o al connec ion
Ω
in oduces addi ional phase e m, namely
pa allel anspo phase
RΩ(˙γ) dτ
on pa h. In pa h in eg al, his e m is equi alen
o adi ional ec o po en ialscala po en ial e ms.
3. Ex emizing ac ion yields gene alized geodesic equa ion, ex a connec ion cu a u e
e ms p oduce o ce-like e ms, espec i ely co esponding o space ime cu a u e
(g a i y) and in e nal eld s eng h (gauge o ce), as well as en opic o ce con i-
bu ion om esolu ion ow.
4. Taking Eh en es limi o wa e packe cen e ajec o y, ope a o expec a ion alue
e olu ion can be simplied o classical equa ion, ob aining o m in p oposi ion.
De ails in Appendix C.
4.3 P oo Ou line o Theo em 3.7
Using equency conse a ion and local ene gy edshi ela ion in s a ic space ime:
1. In s a ic space ime wi h o m
g=−N2d 2+hijdxidxj
, ime Killing ec o
∂
ensu es
ene gy conse a ion.
2. Ene gy measu ed in local ine ial ame is
ωloc =N−1(x)ω
. S a e densi y die -
ence and g oup delay be ween sca e ing egion in e io and dis an egion mus be
exp essed using local ene gy.
3. Unde assump ion ha scale densi y is only escaled by
N(x)
, ime scales a die -
en posi ions die only by posi ion-dependen cons an ac o , i.e., belong o same
equi alence class. Thei a io is
N(x)
in uni a y ans o ma ion.
4. Reconciling his ela ion wi h g a i a ional edshi expe imen al da a can e i y
consis ency o his in e p e a ion.
De ailed dimensional econcilia ion in Appendix B.
4.4 P oo Ou line o Theo em 3.10
This heo em abs ac s unied s uc u e o en opy ex emali ymodula ene gygeome ic
equa ions:
1. Wo k in holog aphy and algeb aic quan um eld heo y shows ha unde app o-
p ia e ci cums ances, modula ow can be geome ically ealized as ow o some
Killing o app oxima e Killing on g a i y side; expec a ion alue o modula Hamil-
onian is ela ed o changes in gene alized en opy.
9
I adop ing opposi e sign con en ion when dening spec al shi unc ion, his minus
sign can be elimina ed; he e we adop con en ion in main ex .
A.3 Wigne Smi h Ma ix and T ace Fo mula
Wigne Smi h ma ix is dened as
Q(ω) = −iS(ω)†∂ωS(ω)
No e ha o any in e ible ma ix
S(ω)
,
∂ωln de S(ω) = [S−1(ω)∂ωS(ω)]
Fo uni a y ma ix
S(ω)
,
S−1(ω) = S†(ω)
, hus
−i∂ωln de S(ω) = −i [S†(ω)∂ωS(ω)] = Q(ω)
On he o he hand,
ln de S(ω) = iΦ(ω)
hus
Q(ω) = −i∂ωln de S(ω)=Φ′(ω) = 2φ′(ω)
Combining abo e esul s, ob ain
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
A his poin , scale iden i y is igo ously es ablished.

B S a ic Space ime, Redshi , and Scale Equi alence
B.1 S a ic Me ic and Local Ene gy
Le s a ic space ime me ic be
g=−N2(x)d 2+hij(x)dxidxj
Killing ec o
ξµ= (∂ )µ
co esponds o conse ed quan i y
E=−pµξµ=−p
Ene gy measu ed in local ine ial ame is
ωloc =pˆ
0=N−1(x)E
whe e
pˆ
0
is o hogonal ame componen .
The e o e, local s a e densi y and g oup delay in sca e ing egion na u ally unc ion
on
ωloc
.
16

B.2 Scale Densi y and Redshi Fac o
Assume scale densi y sa ises
κ(ω;x) = κloc(ωloc) = κloc(N−1(x)ω)
Dis an
N(∞)=1
. Fo gi en ene gy window, in dis an obse e ime scale, g oup
delay is
∆τ(ω;x) = Z1
2π Q(ω;x) dω
I
κloc
a ies slowly wi hin window, hen
∆τ(ω;x)≈N−1(x)∆τ∞(ω)
The e o e, o wo posi ions
x1,x2
, ime scale a io is
∆τ(ω;x2)
∆τ(ω;x1)≈N−1(x2)
N−1(x1)=N(x1)
N(x2)
This is consis en wi h g a i a ional edshi ela ion
ν2/ν1=N(x1)/N(x2)
, showing
bo h belong o same ime scale equi alence class.

C Semi-classical Limi and Fo ce as Cu a u e P ojec-
ion
C.1 Ac ion wi h Connec ion and Pa h In eg al
Conside ac ion con aining connec ion
S[γ] = Z(pµ˙xµ−H ee(x, p)) dτ+Z⟨χ, Ω( ˙γ)χ⟩dτ
whe e
χ
desc ibes in e nal deg ees o eedom and esolu ion a iables,
Ω
is o al
connec ion.
Pa h in eg al weigh ac o is
eiS[γ]
. Va ying ac ion yields
mD2xµ
Dτ2=qFµν˙xν+ µ
es
whe e
Fµν
is ob ained om in e nal pa cu a u e o connec ion,
µ
es
is gi en by
combina ion o esolu ion ow and en opy g adien .
C.2 Eh en es Theo em and Mac oscopic Fo ce
E olu ion o expec a ion alues o ope a o s
ˆxµ
and
ˆpµ
on Hilbe space sa ises Eh en-
es heo em. Taking expec a ion o Hamil onian con aining connec ion, in na ow wa e
packe app oxima ion, expec a ion o ope a o p oduc can be ac o ized in o p oduc
o expec a ions plus noise co ec ion. Igno ing highe -o de noise yields cen e -o -mass
ajec o y equa ion. A his poin ,  o ce is comple ely de e mined by connec ion cu -
a u e.

17
D Modula Flow, QNEC, and G a i a ional Dynamics
D.1 Modula Flow and Rela i e En opy
Fo local algeb a
A(O)
and s a e
φ
, modula ow
σφ
and modula Hamil onian
Kφ=
−log ∆φ
sa is y
σφ
(A) = eiKφ Ae−iKφ
Rela i e en opy
S(ψ||φ) = (ρψlog ρψ−ρψlog ρφ)
in holog aphic co espondence, has di ec ela ion wi h gene alized en opy and g a i y-
side ene gy.
D.2 QNEC and Local Ene gy Cons ain
QNEC exp ession
⟨Tkk⟩ ≥ 1
2πS′′
ensu es small de o ma ion along null di ec ion has second-o de change o en opy
limi ed by ene gy. This can be iewed as a kind o en opy expansion a ow o ime,
and can be used o p o e ha i gene alized en opy akes ex emum on gi en c oss-
sec ion, co esponding geome y mus sa is y local o m o Eins ein equa ion.
D.3 Alignmen Condi ion o Time Scale Equi alence Class
I modula ow
and geome ic ime
τ
belong o same scale equi alence class, mono onic-
i y o modula Hamil onian and Iye Wald wo ken opy ela ion on g a i y side join ly
de i e local g a i a ional dynamics. Con e sely, i geome y sa ises specic ene gy con-
di ions and gene alized second law, can cons uc s a e and algeb a aligning modula ow
ime wi h geome ic ime, hus comple ing closu e o unied ime scale.
This pa in eg a es mul iple esul s om algeb aic QFT, holog aphy, and g a i a-
ional he modynamics, showing ime scale equi alence class has bo h ope a ional deni-
ion om sca e ingspec al shi and s uc u al deni ion om modula owen opy,
which a e in insically consis en unde app op ia e condi ions.
18